1 / 41

# UNIT II: The Basic Theory - PowerPoint PPT Presentation

UNIT II: The Basic Theory. Zero-sum Games Nonzero-sum Games Nash Equilibrium: Properties and Problems Bargaining Games Review Midterm 3/23. 3/9. Review. Review Terms Counting Strategies Prudent v. Best-Response Strategies Graduate Assignment. Review.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' UNIT II: The Basic Theory' - clive

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

• Zero-sum Games

• Nonzero-sum Games

• Nash Equilibrium: Properties and Problems

• Bargaining Games

• Review

• Midterm 3/23

3/9

• Review Terms

• Counting Strategies

• Prudent v. Best-Response Strategies

Dominant Strategy: A strategy that is best no matter what the opponent(s) choose(s).

Prudent Strategy:A prudent strategy maximizes the minimum payoff a player can get from playing different strategies. Such a strategy is simply maxsmintP(s,t) for player i.

Mixed Strategy:A mixed strategy for player i is a probability distribution over all strategies available to player i.

Best Response Str’gy: A strategy, s’, is a best response strategy iff Pi(s’,t) > Pi(s,t) for all s.

Dominated Strategy: A strategy is dominated if it is never a best response strategy.

Saddlepoint:A set of prudent strategies (one for each player), s. t. (s*, t*) is a saddlepoint, iff maxmin = minmax.

Nash Equilibrium: a set of best response strategies (one for each player), (s’,t’) such that s’ is a best response to t’ and t’ is a b.r. to s’.

Subgame: a part (or subset) of an extensive game, starting at a singleton node (not the initial node) and continuing to payoffs.

Subgame Perfect Nash Equilibrium (SPNE): a NE achieved by strategies that also constitute NE in each subgame. eliminates NE in which the players threats are not credible.

Player 1

Player 2 has 4 strategies:

Left Right

L R L R

-2 4 2 -1

Player 2

LL RR LR RL

-2 4 -2 4

2 -1 -1 2

L

R

GAME 2: Button-Button

Player 1

Player 2 has 4 strategies:

Left Right

L R L R

-2 4 2 -1

Player 2

LL RR LR RL

-2 4 -2 4

2 -1 -1 2

L

R

GAME 2: Button-Button

Player 1

Player 2 has 4 strategies:

Left Right

L R L R

-2 4 2 -1

Player 2

LL RR LR RL

-2 4 -2 4

2 -1 -1 2

L

R

GAME 2: Button-Button

Player 1

Player 2 has 4 strategies:

Left Right

L R L R

-242 -1

Player 2

LL RR LR RL

-2 4 -2 4

2 -1 -1 2

L

R

GAME 2: Button-Button

Player 1

If Player 2 cannot observe Player 1’s choice …

Player 2 will have fewer strategies.

Left Right

L R L R

-2 4 2 -1

Player 2

GAME 2: Button-Button

L R

L

R

Player 1

-2 4

2 -1

Left Right

L R L R

-2 4 2 -1

Player 2

GAME 2: Button-Button

Battle of the Sexes

O F

O

F

Player 1

2, 1 0, 0

0, 0 1, 2

Opera Fight

O F O F

(2,1) (0,0) (0,0) (1,2)

Player 2

Find all the NE of the game.

Battle of the Sexes

O F

O

F

Player 1

2, 1 0, 0

0, 0 1, 2

Opera Fight

O F O F

(2,1) (0,0) (0,0) (1,2)

Player 2

NE = {(1, 1); (0, 0); }

Battle of the Sexes

O F

O

F

Player 1

2, 1 0, 0

0, 0 1, 2

Opera Fight

O F O F

(2,1) (0,0) (0,0) (1,2)

Player 2

NE = {(O,O); (F,F); }

Prudence v. Best Response

O F

P2

2

1

2, 1 0, 0

0, 0 1, 2

NE (0,0)

O

F

NE (1,1)

1 2

P1

NE = {(1, 1); (0, 0); (MNE)}

Mixed Nash Equilibrium

Prudence v. Best Response

OF

Let (p,1-p) = prob1(O, F)

(q,1-q) = prob2(O, F)

2, 1 0, 0

0, 0 1, 2

O

F

NE = {(1, 1); (0, 0); (MNE)}

Prudence v. Best Response

O F

Let (p,1-p) = prob1(O, F)

(q,1-q) = prob2(O, F)

Then

EP1(Olq) = 2q

EP1(Flq) = 1-1q

EP2(Olp) = 1p

EP2(Flp)= 2-2p

2, 1 0, 0

0, 0 1, 2

O

F

NE = {(1, 1); (0, 0); (MNE)}

Prudence v. Best Response

O F

Let (p,1-p) = prob1(O, F)

(q,1-q) = prob2(O, F)

Then

EP1(Olq) = 2q

EP1(Flq) = 1-1q

q* = 1/3

EP2(Olp) = 1p

EP2(Flp)= 2-2p

p* = 2/3

2, 1 0, 0

0, 0 1, 2

O

F

NE = {(1, 1); (0, 0); (2/3,1/3)}

}

Prudence v. Best Response

q=1 q=0

Player 1’s Expected Payoff

EP1

2/3

0

2

p=1

2,1 0,0

0, 0 1, 2

p=1

p=0

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Prudence v. Best Response

q=1 q=0

Player 1’s Expected Payoff

EP1

2/3

0

2

EP1 = 2q +0(1-q)

2,1 0,0

0, 0 1, 2

p=1

p=0

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Prudence v. Best Response

q=1 q=0

Player 1’s Expected Payoff

EP1

1

2/3

0

2

0

p=1

2,1 0,0

0, 0 1, 2

p=1

p=0

p=0

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Prudence v. Best Response

q=1 q=0

Player 1’s Expected Payoff

EP1

1

2/3

0

2

0

p=1

2,1 0,0

0, 0 1, 2

p=1

p=0

EP1 = 0q+1(1-q)

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Prudence v. Best Response

q=1 q=0

Player 1’s Expected Payoff

EP1

1

2/3

0

2

0

Opera

2,1 0,0

0, 0 1, 2

p=1

p=0

Fight

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Prudence v. Best Response

q=1 q=0

Player 1’s Expected Payoff

EP1

1

2/3

0

2

0

2,1 0,0

0, 0 1, 2

p=1

p=0

p=1

0<p<1

p=0

0<p<1

q = 1/3

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Prudence v. Best Response

If Player 1 uses her (mixed) b-r strategy (p=2/3), her expected payoff varies from 1/3 to 4/3.

q=1 q=0

EP1

2/3

1/3

2

0

2,1 0,0

0, 0 1, 2

p = 2/3

p=1

p=0

4/3

p=1

p=0

q = 1/3

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Prudence v. Best Response

If Player 2 uses his (mixed) b-r strategy (q=1/3), the expected payoff to Player 1 is 2/3, for all p.

q=1 q=0

EP1

2/3

1/3

2/3

2

0

2,1 0,0

0, 0 1, 2

p=1

p=0

4/3

p=1

p=0

q = 1/3

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Prudence v. Best Response

O F

Find the prudent strategy for each player.

2, 1 0, 0

0, 0 1, 2

O

F

Prudence v. Best Response

O F

Let (p,1-p) = prob1(O, F)

(q,1-q) = prob2(O, F)

Then

EP1(Olp) = 2p

EP1(Flp) = 1-1p

p* = 1/3

EP2(Oiq) = 1q

EP2(Flq)= 2-2q

q* = 2/3

2, 1 0, 0

0, 0 1, 2

O

F

Prudent strategies: 1/3; 2/3

Prudence v. Best Response

O F

Let (p,1-p) = prob1(O, F)

(q,1-q) = prob2(O, F)

Then

EP1(Olq) = 2q

EP1(Flq) = 1-1q

q* = 1/3

EP2(Olp) = 1p

EP2(Flp)= 2-2p

p* = 2/3

2, 1 0, 0

0, 0 1, 2

O

F

NE = {(1, 1); (0, 0); (2/3,1/3)}

Prudence v. Best Response

If Player 1 uses her prudent strategy (p=1/3), expected payoff is 2/3 no matter what player 2 does

O F

EP1

2/3

1/3

2

0

2, 1 0, 0

0, 0 1, 2

p = 2/3

O

F

4/3

p=1

p=0

p = 1/3

q

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)}

Prudence v. Best Response

If both players use (mixed) b-r strategies, expected payoff is 2/3 for each.

O F

EP1

2/3

1/3

2

0

2, 1 0, 0

0, 0 1, 2

p = 2/3

O

F

4/3

p=1

p=0

p = 1/3

q = 1/3

q

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)}

Prudence v. Best Response

If both players use (mixed) b-r strategies, expected payoff is 2/3 for each.

O F

P2

2

1

2/3

2, 1 0, 0

0, 0 1, 2

NE (0,0)

O

F

NE (1,1)

2/31 2

P1

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)}

Prudence v. Best Response

If both players use prudent strategies, expected payoff is 2/3 for each.

O F

P2

2

1

2/3

2, 1 0, 0

0, 0 1, 2

NE (0,0)

O

F

NE (1,1)

2/31 2

P1

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent: {(1/3, 2/3)}

Prudence v. Best Response

BATNA: Best Alternative to a Negotiated Agreement

O F

P2

2

1

2/3

2, 1 0, 0

0, 0 1, 2

NE (0,0)

O

F

BATNA

NE (1,1)

2/31 2

P1

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent: {(1/3, 2/3)}

Prudence v. Best Response

O F

P2

2

1

2/3

2, 1 0, 0

0, 0 1, 2

NE (0,0)

O

F

NE (1,1)

2/31 2

P1

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)}

Is the pair of prudent strategies an equilibrium?

Prudence v. Best Response

NO: Player 1’s best response to Player 2’s prudent strategy (q=2/3) is Opera (p=1).

O F

EP1

2/3

1/3

2/3

2

0

Opera

2, 1 0, 0

0, 0 1, 2

p = 2/3

O

F

4/3

p=1

p=0

p = 1/3

q = 1/32/3

q

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)}

[I]f game theory is to provide a […] solution to a game-theoretic problem then the solution must be a Nash equilibrium in the following sense. Suppose that game theory makes a unique prediction about the strategy each player will choose. In order for this prediction to be correct, it is necessary that each player be willing to choose the strategy predicted by the theory. Thus each player’s predicted strategy must be that player’s best response to the predicted strategies of the other players. Such a prediction could be called strategically stable or self-enforcing, because no single player wants to deviate from his or her predicted strategy (Gibbons: 8).

STABILITY: Is it self-enforcing?

YES YES

UNIQUENESS: Does it identify an unambiguous course of action?

YES NO

EFFICIENCY: Is it at least as good as any other outcome for all players?

--- (YES) NOT ALWAYS

SECURITY: Does it ensure a minimum payoff?

YES NO

EXISTENCE: Does a solution always exist for the class of games? YES YES

Problems of Nash Equilibrium

• Indeterminacy: Nash equilibria are not usually unique.

2. Inefficiency: Even when they are unique, NE are not always efficient.

Problems of Nash Equilibrium

• T1 T2

Multiple and Inefficient Nash Equilibria

S1

S2

5,5 0,1

1,0 3,3

When is it advisable to play a prudent strategy in a nonzero-sum game?

Problems of Nash Equilibrium

• T1 T2

Multiple and Inefficient Nash Equilibria

S1

S2

5,5 -99,1

1,-99 3,3

When is it advisable to play a prudent strategy in a nonzero-sum game?

What do we need to know/believe about the other player?

Bargaining games are fundamental to understanding the price determination mechanism in “small” markets.

The central issue in all bargaining games is credibility: the strategic use of threats, bluffs, and promises.

When information is asymmetric, profitable exchanges may be “left on the table.”

In such cases, there is an incentive to make oneself credible (e.g., appraisals; audits; “reputable” agents; brand names; lemons laws; “corporate governance”).